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Abstract

The control of highly infectious diseases of livestock such as classical swine fever,
foot-and-mouth disease, and avian influenza is fraught with ethical, economic, and
public health dilemmas. Attempts to control outbreaks of these pathogens rely on massive
culling of infected farms, and farms deemed to be at risk of infection. Conventional
approaches usually involve the preventive culling of all farms within a certain radius
of an infected farm. Here we propose a novel culling strategy that is based on the
idea that farms that have the highest expected number of secondary infections should
be culled first. We show that, in comparison with conventional approaches (ring culling),
our new method of risk based culling can reduce the total number of farms that need
to be culled, the number of culled infected farms (and thus the expected number of
human infections in case of a zoonosis), and the duration of the epidemic. Our novel
risk based culling strategy requires three pieces of information, viz. the location of all farms in the area at risk, the moments when infected farms are
detected, and an estimate of the distance-dependent probability of transmission.

Introduction

Epidemics of infectious diseases such as classical swine fever, food-and-mouth, and
avian influenza continue to wreak havoc in commercial livestock [1-6]. Efforts to control such outbreaks rely heavily on culling of infected farms, and
farms in the vicinity of infected farms. This approach induces massive economic costs
and leads to great animal suffering. It is therefore desirable to make as efficient
as possible use of the available resources, and to spare as many animals as possible.
Furthermore, in case of diseases that have zoonotic potential, such as highly pathogenic
avian influenza A viruses of the H5 and H7 subtypes, it is also important to minimize
the risk of human exposure [7-11].

The aim of preventive culling in outbreaks of commercial livestock is to contain the
epidemic by removing susceptible flocks in the vicinity of infected farms; a typical
strategy used for this is ring culling. In this strategy all farms within a certain
radius of an infected farm are culled, typically starting close to the infected farm(s).
The distance to an infected farm is related to the probability of a farm becoming
infected. Ring culling therefore essentially involves culling the farms with the highest
probability of becoming infected. We argue that, in addition to the distance to infected
farms, another factor that is important is the local density of neighbouring farms.
The local density determines how an epidemic is likely to develop, e.g.: a farm in
an area with high density will likely cause more new infections than a farm in area
with a low density; this implicitly follows from the relationship between distance
and risk. In previous work the number of new infections that each infected farm is
expected to cause was quantified by a farm reproduction number (R) and they were used
to create risk maps that indicated areas with potential for high epidemic spread [12].

In this paper a novel culling strategy is introduced that takes into account not only
the distance of susceptible farms to the infected farms, but also the number of secondary
infections that a susceptible farm is expected to produce should it become infected.
Specifically, we calculated for each farm not yet (known to be) infected a so-called
risk value, which represents the number of infections the farm is expected to produce
given current information on the unfolding of the epidemic. We argue that farms which
rank highest in the risk based ordering should be culled first, thereby achieving
an efficient allocation of resources (i.e. time, money, equipment). In practice, the
risk value of each susceptible farm is given by the probability that a farm will become
infected in a certain time span multiplied by the reproduction number of the farm
once it is infected. Similar ideas that incorporate the connectivity of farms or individuals
in applying an intervention measure have been suggested before in (non-spatial) network
models of infectious disease spread [13]. For example by vaccinating friends' of friends which proved to be more effective
than vaccinating people at random [14]. Similar is also the strategy to vaccinate children with the aim to reduce disease
transmission [15] or to preferentially vaccinate large urban centres to reduce their role as disease
reservoirs [16].

Our risk based culling scheme works with three pieces of information. First the locations
of all farms in the area at risk need to be known. Second, an assessment of the current
state of the epidemic should be available, in particular which farms are infected,
and which farms are still susceptible. Third, an estimate of how the transmission
probability depends on the distance between infected and susceptible farms should
be at hand. The first two pieces are usually readily available during an epidemic.
For the third piece of information estimates from past epidemics can usually be used
[1,4,12,17].

We evaluate the performance of the risk based culling strategy in a simulation study
that is loosely based on a large outbreak of highly pathogenic H7N7 avian influenza
in the Netherlands. Parameter values and the transmission hazard are based on experience
with this outbreak [6,12]. Throughout we systematically compare the effectiveness of our risk based culling
strategy with the traditional approach that relies on the culling of farms in a ring
(1-3 km) around infected premises. The comparisons are based on (1) the number of
infected farms culled, which is related to the expected number of human infections,
(2) the duration of the epidemic, and (3), the total number of farms culled.

Materials and methods

Modelling

The spread between farms was modelled with a stochastic SEIR model (Susceptible-Exposed-Infected-Removed)
that operates with fixed time steps of 1 day. The probability qi that a susceptible farm i is infected on a particular day t is given by:

(1)

where λi(t), the force of infection on farm i at day t, is calculated according to:

(2)

The function h(rij) is called the transmission kernel. It is defined as the infection hazard posed by
farm j to farm i as a function of the inter-farm distance rij [12], 1 is an indicator function that is 1 if j is infected (Figure 1) and 0 otherwise. In equation 2 it is assumed that transmission between farms is
distance dependent, which for many outbreaks provided a satisfactory description of
the data [1,4,12,17]. A number of mechanisms may be able to cause spread from farm to farm. Virus may
for example be carried over by people, vehicles, or wind. In our model we adopt a
phenomenological approach, and do not explicitly model different transmission mechanisms.

Figure 1.Classification of the farms in the model. Farms are either susceptible to infection, infected but not yet infectious (exposed),
infected and infectious, detected and not infectious anymore, or removed from the
system by culling.

Although it is our goal to investigate the efficiency of risk based culling strategies
in general, the parameters are specifically tailored to mimic the spatial spread of
highly pathogenic avian influenza viruses in densely populated poultry areas. The
shape of the hazard and estimates of the parameters to scale the hazard were estimated
from the outbreak of highly pathogenic H7N7 avian influenza in the Netherlands [12]. The shape that fitted best was of the form:

(3)

where h0, r0, and α are parameters of the hazard function (Table 1), and r is the distance between farms.

Table 1. Settings of the hazard kernel (equation 3) as used in the base scenario and the scenarios
of the sensitivity analyses

We assume that upon infection each farm first becomes exposed (i.e. infected but not
yet infectious) for a period of two days (Figure 1) [12,18]. After the exposed period has elapsed the farm is assumed to be infectious until
it is detected. Here we assume for simplicity that upon detection the farm immediately
ceases to be infectious to other farms, e.g., due to appropriate biosafety measures.
In our simulation the time between becoming infectious to becoming detected is drawn
from a gamma distribution with a mean (T) of 7 days and a shape parameter (c) of 100 (Figures 1 and 2a). Culling in the model is done (as in reality) with limited daily capacity. Farms
that are culled are removed from the system, and cease to play a role in the infection
dynamics. The culling capacity is assumed to be low at start of the epidemic and then
increases quickly reaching a maximum after 11 days (Figure 2b) and is based on the situation of 2003 [6]. We assume that culling of detected infected farms is given priority above preventive
culling. Each time step all detected infected farms are therefore culled first providing
capacity is available (Figure 1 and Figure 2b). If the number of infected farms is larger than the culling capacity on a particular
day then the infected farms detected last are culled the next day(s). If the culling
capacity is greater than the number of detected infected farms then the remaining
culling capacity is used for preventive culling. Preventively culled farms, whether
they were susceptible, exposed or infected, become removed (Figure 1).

Figure 2.Overview of model parameters. (a) Distribution of the days to detection of an infected farm, (b) Culling capacity
as a function of the time since detection of the outbreak, (c) Hazard kernels with
an increased and decreased tail, and the misspecified kernel, (d) Hazard kernels with
increased and decreased capacity.

Culling strategies

We now define in more detail the two culling strategies considered in this paper:
ring culling and risk based culling. With ring culling all farms within a certain
radius of any of the farms where an infection was detected are preventively culled.
Culling is then continued until there are no more farms present in any of the (possibly
overlapping) rings around infected farms. Ring culling is typically carried out inside-out,
i.e. starting near the infected farm on the inside of the ring. In our model this
is mimicked by consistently selecting the farm that is closest to a farm where an
infection has been detected. An alternative ring culling strategy works from the outside
of the ring to the inside. The rationale for this strategy is that it may help contain
the infection within the culling ring. In this strategy the farm that is furthest
away (but within the ring) of any farm where an infection was detected is culled first.
In The Netherlands in 2003 culling was started on the inside of the ring close to
the infected farms. At the start of the epidemic a 1 km ring was used, and later in
the epidemic 3 km rings were used. In our analysis we consider both these radii. In
our calculations for outside-in ring culling we considered a scenario with a ring
radius of 3 km.

With risk based culling the estimated number of infections each farm is expected to
create is used as culling criterion. The candidate farm with the highest expected
number of new infections is then preventively culled first. The expected number of
infections Ei for farm i is given by the product of the probability that farm i is infected (qi, equation 1) and its reproduction number (Ri, equation 7):

(6)

Assuming a gamma distributed infectious period with mean T and a shape parameter (c) of 100, the reproduction value (Ri) is calculated for each remaining farm (not detected, not culled), only taking into
account all other remaining farms, as follows [12]:

(7)

where i and j run through all farms that are not detected and that not have been culled.

The exact probability for a farm to become infected (qi, equation 1) can in practise not be calculated during the epidemic as this would
require complete knowledge on which farms are currently infected (also the ones that
are yet undetected). However, it is possible to approximately calculate the risk that
farms were infected in the past based on knowledge of the detected infections. In
this calculation, we first assume that all detected farms have been infectious for
exactly T days (mean infectious period, seven days). We note that this assumption could be dropped
if detailed knowledge on the infectious period of the source were available during
the epidemic. Then we approximate how much exposure each farm has had in the past
T days. Suppose one particular farm j is detected as infected at time tjd. This means that all neighbouring farms have been exposed to this infected farm for
the previous T days. If on one particular neighbouring farm no infection is detected up until x days after the detection of its infected neighbour (with x smaller than or equal to T) it should have escaped infection by that infected neighbour for at least T-x days (Figure 3). In agreement with this we estimate an approximate cumulative hazard λi* of infection for each farm i where no infection was detected up until time t according to:

Figure 3.Overview of how to calculate the approximate risk that farms were infected in the
past based on knowledge of the detected infections (equation 8). In the example below, at day 7 an infected farm is identified as being infected.
On day 7 a neighboring farm has on average been exposed to the infected farm for the
last 7 days. If the susceptible farm was not detected as being infected then on day
8, it has been on average exposed for 6 days. And so on, until day 14 when on average
there is no further exposure. In equation 5 at, for example, day 9, t = 9, tjd = 7, and T = 7 which results in 5 days exposure.

(8)

The hazard is approximate (as indicated with the *) because farms that are infected
but have not been yet been detected are not taken into account because these are unknown
at that point in time. The approximate probability (qi*) that farm i was infected in the past based can be calculated according to

(9)

This probability is used to calculate the approximate expected number of infections
caused by farm i:

(10)

is used to rank the farms according to risk in order to determine the order for risk
based preventive culling. In our model we used a risk threshold ("thr") below which
farms are not culled anymore. In our model calculations we analyze the sensitivity
of our results to changes in the threshold value.

Simulation details

The efficiency of various conventional ring culling strategies and the novel risk
based culling strategy were assessed using simulations of outbreaks on maps with randomly
generated farm locations (see Additional file 1, Supplementary text and Additional file 2, Figure S1). Each map consisted of a circular inner area of 1000 farms with a density
of 3.8 farms per m2 and an outer (ring-shaped) area of 1000 farms with a density of
0.5 farms per m2. The density assumed in the inner area is equal to the density in
the largest poultry area of the Netherlands which is where most of the 2003 epidemic
of avian influenza occurred. The density of the outer area is the density observed
in the remainder of The Netherlands.

To prevent early stochastic fade-out, and to condition on a large epidemic, all simulations
were seeded with 10 infectious and 10 exposed farms (Figure 4). The starting configuration of the 10 infectious and 10 exposed farms was itself
created with the transmission model (equations 1-3 and Figure 1). The first infected farm was randomly selected. The start-up simulation was run
until 10 farms were infectious and 10 farms were exposed. In this manner we generated
50 different random starting configurations. For each culling scenario considered
we did 400 simulations in which we used each map 8 times. For base-line values for
the kernel parameters (Table 1) we carried out 2000 additional simulations (40 per map) for three of the culling
strategies to evaluate the performance of these culling strategies per map.

Figure 4.Examples of outbreaks (a) homogeneous Poisson clustered, (b) moderately clustered
(base scenario), and (c) clustered map. Left graph shows a starting position with infected farms in red, the right graph
shows an end situation with culled farms in blue.

In a future epidemic the exact shape of the hazard kernel of the disease spread is
likely to be unknown. We therefore tested the sensitivity of the results to changes
in the hazard kernel (Table 1), while the kernel to calculate the risk as a basis for preventive culling was left
unchanged. Specifically, we considered two additional scenarios, one with a heavy-tailed
kernel, and one in which the tail of the kernel falls of more quickly than in the
default scenario (Figure 2d). We also investigated a situation with a more extensive difference between the hazard
kernel used to calculate the disease risk and the actual hazard kernel of the disease
spread (i.e. a misspecified kernel). For this we used a kernel that has equal disease
spread disease up to 10 km (Figure 2c). The hazard kernels were adjusted so that the average reproduction number across
all maps was the same for all three scenarios (i.e. R = 1.65) (Table 1). We furthermore considered scenarios with a high (R = 2.1) and low (R = 1.2) reproduction number (Table 1 and Figure 2d). Finally, in the base scenario the maximum culling capacity was fixed at 24 farms
per day. To investigate the robustness of our results to the exact culling capacity
we investigated scenarios with an increased (32 farms per day) or decreased (16 farms
per day) culling capacity.

The relevant outputs of the model are the total number of farms culled, the number
of infectious farms culled, and the duration of the epidemic (defined as the time
from the first detection until the last culling of an infected farm). The total number
of farms that are culled and the duration of the epidemic are both measures with economic
relevance, not only because of the direct costs of culling, but also because under
EU regulations borders will be closed for export during (an for some period after)
an epidemic of avian influenza. Furthermore, minimizing animal suffering in itself
is a worthy goal. The number of farms that are culled while being infectious is relevant
because it determines the level of human exposure to an agent with zoonotic potential
[7-11].

The simulation results obtained contain three sources of variation, i.e. (1) the culling
strategy, (2) the random maps, and (3) the stochastic epidemic process. To separate
these variances, and to single out the effect of the culling strategy, we analyzed
the simulation results with a linear mixed model that used the maps as a random effect
and the culling strategy as a fixed effect. The mixed model was used to estimate confidence
bounds for the effect of the various culling strategies.

Results

Risk based culling reduced the number of infected farms culled compared to both 1
and 3 km ring culling strategies, and thereby should be able to reduce the number
of human infections (Table 2). Of three criteria by which we tested the control strategies, (1) number of infected
farms culled, (2) total number of farms culled, and (3) duration of the epidemic,
risk based culling outperformed ring culling always on two out of three criteria.
On the third criterion it typically performs either equally well or slightly better
(Tables 2 and 3). Risk based culling has a much lower number of infected and total number of farms
culled compared to 3 km ring culling, and usually shortens the length of the epidemic
(depending on the risk threshold). Compared to 1 km ring culling, risk based culling
has a lower number of infected farms culled, and the duration of the epidemic is shorter.
The total number of farms culled is either equal or lower (depending on the risk threshold).

Table 2. Simulation results for the various risk based and ring culling strategies in the base
scenario

Table 3. Overview of results obtained with various scenarios used in the sensitivity analysis

There was substantial variation between the maps. Per map the relative improvement
of risk based culling over ring culling remained approximately the same. The mean
number of infected farms per map culled (across the three main strategies) ranged
from 82 to 474, the total number of farms culled ranged from 956 to 1423, and the
duration of the epidemic ranged from 53 to 64. The performance of each strategy per
map was studied by carrying out an additional 2000 simulations per strategy, i.e.
40 for each map. On 6 out of 50 maps 1 km ring culling had a lower total number of
farms culled than risk based culling, and 3 km ring culling gave a shorter epidemic
on 5 maps. With the exception of these, risk based culling consistently outperformed
1 km and 3 km ring culling.

In the sensitivity analyses we investigated the effect of changes in culling capacity,
reproduction number, tail of the hazard kernel, the level of clustering of farms,
and a misspecified hazard kernel (Table 3). Qualitatively, the results remained the same in all sensitivity analyses. In all
scenarios risk based culling outperformed ring culling on two or three criteria. Quantitatively,
the total or infected number of farms culled, and the length of the epidemic varied
greatly between the scenarios. The relative improvement of risk based culling over
ring culling, however, remained approximately the same.

Discussion

Preventive culling of farms is an important control measure to halt epidemics of highly
infectious diseases of livestock such as classical swine fever, foot-and-mouth disease,
and avian influenza. This paper introduces a novel prioritization scheme for culling
of farms that is based on the idea that farms with the highest expected number of
secondary infections should be culled first. Our simulations show that risk based
culling outperforms ring culling in terms of the number of infected farms culled,
the total number of farms culled, and the duration of the epidemic. As risk based
culling reduced the number of infected farms that are culled it is therefore also
expected to reduce the number of human infections. We find substantial variation in
the outcome between different maps but for a given map risk based culling consistently
outperformed ring culling. This indicates that the spatial structure has a large influence
on the outcome of an epidemic, which is supported by previous research [4,17,19-21].

Although the model presented here is parameterised for the avian influenza epidemic
that occurred in The Netherlands in 2003, the methodology of risk based culling is
more generally applicable to other infectious diseases controlled by culling. The
only information needed are the locations of the farms, the moments at which infected
farms were culled (both essential for any control and usually available from surveillance),
and an estimate of the distance-dependent probability of transmission. An extension
of our method that could potentially further improve risk based culling would be to
not only focus on the expected number of infections within one infection generation,
but try to estimate the expected number of infections in second and perhaps even third
infection generations in the future.

We assumed that all farms are equally infectious which is reasonable for avian influenza
[12] but for other diseases this may be different. Variability in susceptibility and infectivity
can also be taken into account in risk based culling, proving estimates are available.
One example where such variability existed is foot-and-mouth outbreak in the UK. For
this epidemic a model was derived that is similar to the model used in this paper
[3,4,17] and it can be used to calculate the probability of infection and reproduction number
per farm as needed for risk based culling.

It is possible with an extensive misspecification in, for example, the infectivity
of farms that risk based culling would be less effective. Note though that alternative
strategies (such as ring culling) may suffer similarly. The challenge here is to have
a good epidemiological understanding of how a disease spreads and incorporate this
knowledge into the calculations. We believe that if misspecifications are minor, the
reproductive number still identifies patches of farms that are close together weighted
by their distance to infected farms. Quantitatively the outcomes may differ to some
extent but qualitatively (risk based culling is about prioritizing) they may still
be accurate. The effectiveness of risk based culling also depends on the culling capacity
relative to the spread of disease. If the culling capacity is too low any control
is impossible. If the culling capacity is very high then the order of culling becomes
irrelevant. In between these extremes, culling resources need to be used efficiently
and risk based culling can aid in this.

One advantage of risk based culling is that it does not require a certain arbitrary
ring to be set. It can be argued however that the threshold needed in risk based culling
to set the minimum risk level for culling is also arbitrary, and there is indeed not
one clear risk based threshold (thr) that achieves the best results across all three criteria. If the threshold is decreased,
the number of farms culled is increased and the pool of susceptible farms depletes
quicker, which means an epidemic is stopped earlier. An epidemic that stops more rapidly
is likely going to have less infected farms culled, and thus less human infections.
Vice versa, if the threshold is increased, the total number of farms culled decreases
but the length of the epidemic and the number of infected farms culled increase. Which
strategy is best thus depends what goals decision makers want to achieve. Economically
the cost of the total number of farms culled and the cost of a longer epidemic can
be weighted. The impact on public health (human casualties) is however difficult to
weigh and are dependent on the disease. With a disease like avian influenza which
has a clear zoonotic potential reducing the public health impact is arguably the most
important.

For policy makers our risk based culling policy may be more difficult to justify to
stake holders and the public than the simple traditional ring culling strategy. In
addition, to be acceptable any culling strategy would have to satisfy the requirements
of regulatory bodies. An intuitively appealing strategy may be to apply a risk based
prioritization scheme within a culling ring. In our results this proved to be quite
efficient (as shown in Table 2), primarily because most farms that are selected in a risk based prioritization scheme
are located within a ring of 3 km from an infected farm (Table 2).

In the past mainly ring culling strategies have been considered in practice and literature
[3,4,20,22]. In [20,22] a strategy was modelled that prioritised farms with high probability of infection.
In this work the probability of infection per farm was based on the distance to the
infected farms weighted by the number of sheep and cows. In risk based culling selection
of farms is done by combining the probability of infection (dependent on distance
to the infected farms) with the reproduction number (dependent on the local density
of farms). Our results demonstrate that including the local density of farms to determine
the order of preventive culling to control an epidemic is a promising strategy. This
paper provides a guideline that could help improve the effectiveness of culling.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

DB, JS, MK, MB conceived the study, TH, MB, DB formulated the model equations, DB
programmed the model, carried out the simulations, and drafted the manuscript. All
authors read, amended and approved the final manuscript.

Acknowledgements

This work was supported by CDC under the grant agreement U19 CI000404-01, Avian influenza
collaborative research centers, Studies at the human animal interface, by the "Impulse
Veterinary Avian Influenza Research in The Netherlands" program of the Dutch government,
and by the Strategic Research theme Infectious Disease Dynamics of the Dutch National
Institute of Public Health and the Environment. We would like to thank Jacco Wallinga
for his input, and Jan van de Kassteele for his statistical advice.

References

Stegeman A, Elbers AR, Smak J, de Jong MC: Quantification of the transmission of classical swine fever virus between herds during
the 1997-1998 epidemic in The Netherlands.